Cohomology theories as colimits I am looking for examples of cohomology theories that can be written as (filtered, or another nice class of) colimits of "simpler" functors, i.e. which $\{h^n : {\bf Top}^2 \to {\bf Ab}\}_n$ are such that
$$
h^n(X) \cong \text{colim}_j\; h_j^n(X)
$$
for a suitable diagram ${\cal J}\to [{\bf Top}^2,{\bf Ab}]$. 
Of course this is a really vague question:


*

*A "cohomology theory" (and the category thereof) is what (for example) Rudyak I.3.8 defines as such.

*I'm not asking that the $h^n_j$ are cohomology theories themselves, but you can assume this additional requirement.

*You're quite free to interpret the word "simple" in the way to like more. I'm in fact explicitly asking for which meaning of "simple" this question has a good answer.


The question remains a bit vague: whatever $h^n(X)$ is, you can take a presentation for this abelian group and say that it is a colimit. Nevertheless I think that asking for a colimit of functors is a bit more restrictive and avoids trivial cases.
My feeling is that the answer is always "quasi-affirmative": a cohomology theory, i.e. a spectrum, belongs to a presentable quasicategory. But spectra and cohomology theories aren't really the same thing.
 A: Suppose you just look at colimits indexed by $\mathbb{N}$, and fix the cohomological degree $n$.  I claim that there must exist $j$ such that the map $h_j^n(X)\to h^n(X)$ is surjective for all $X$.  Indeed, if not, we can choose spaces $X_j$ and classes $a_j\in h^n(X_j)$ that do not come from $h^n_j(X_j)$.  Put $X=\coprod_jX_j$.  As $h$ is assumed to be a cohomology theory, there is a unique element $a\in h^n(X)$ with $a|_{X_j}=a_j$ for all $j$.  If this comes from some $a'\in h^n_j(X)$ then we see that $a'|_{X_j}$ maps to $a_j$, contrary to assumption.  (We do not need to assume that $h_j^n$ converts coproducts to products for this, we are just using functoriality.)  
I suspect that under mild assumptions one can prove something stronger: there is a cofinal sequence $j_1<j_2<\dotsb$ and functors $u_k$ such that $h_{j_k}^n=h^n\oplus u_k$ with the map $h_{j_k}\to h_{j_{k+1}}$ being zero on $u_k$.  This means that there are no really interesting examples.  All this depends on the colimit formula being valid for infinite complexes, of course.
A: I think by restriction to finite complexes of the source of a given map $f:X\to BO$,  one obtains a family of spectra which filter Thom spectrum of $f$. As famous examples, we may consider Ravenel spectra $X(n)$ as well as various Madsen-Tillmann spectra such as $MTO(n)$. Of course, I don't claim this gives all possible examles. This is perhaps dual to what you ask for.
